Abstract
We propose an analytic method to calculate the matter field Kähler metric in heterotic compactifications on smooth Calabi-Yau three-folds with Abelian internal gauge fields. The matter field Kähler metric determines the normalisations of the \( \mathcal{N} \) = 1 chiral superfields, which enter the computation of the physical Yukawa couplings. We first derive the general formula for this Kähler metric by a dimensional reduction of the relevant supergravity theory and find that its T-moduli dependence can be determined in general. It turns out that, due to large internal gauge flux, the remaining integrals localise around certain points on the compactification manifold and can, hence, be calculated approximately without precise knowledge of the Ricci-flat Calabi-Yau metric. In a final step, we show how this local result can be expressed in terms of the global moduli of the Calabi-Yau manifold. The method is illustrated for the family of Calabi-Yau hypersurfaces embedded in \( {\mathrm{\mathbb{P}}}^1\times {\mathrm{\mathbb{P}}}^3 \) and we obtain an explicit result for the matter field Kähler metric in this case.
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P. Candelas, G.T. Horowitz, A. Strominger and E. Witten, Vacuum configurations for superstrings, Nucl. Phys. B 258 (1985) 46 [INSPIRE].
B.R. Greene, K.H. Kirklin, P.J. Miron and G.G. Ross, A three generation superstring model. 1. Compactification and discrete symmetries, Nucl. Phys. B 278 (1986) 667 [INSPIRE].
B.R. Greene, K.H. Kirklin, P.J. Miron and G.G. Ross, A three generation superstring model. 2. Symmetry breaking and the low-energy theory, Nucl. Phys. B 292 (1987) 606 [INSPIRE].
J. Distler and B.R. Greene, Aspects of (2, 0) string compactifications, Nucl. Phys. B 304 (1988) 1 [INSPIRE].
J. Distler and S. Kachru, (0, 2) Landau-Ginzburg theory, Nucl. Phys. B 413 (1994) 213 [hep-th/9309110] [INSPIRE].
S. Kachru, Some three generation (0,2) Calabi-Yau models, Phys. Lett. B 349 (1995) 76 [hep-th/9501131] [INSPIRE].
V. Braun, Y.-H. He, B.A. Ovrut and T. Pantev, A heterotic standard model, Phys. Lett. B 618 (2005) 252 [hep-th/0501070] [INSPIRE].
V. Braun, Y.-H. He, B.A. Ovrut and T. Pantev, A standard model from the E 8 × E 8 heterotic superstring, JHEP 06 (2005) 039 [hep-th/0502155] [INSPIRE].
V. Braun, Y.-H. He, B.A. Ovrut and T. Pantev, The exact MSSM spectrum from string theory, JHEP 05 (2006) 043 [hep-th/0512177] [INSPIRE].
V. Bouchard and R. Donagi, An SU(5) heterotic standard model, Phys. Lett. B 633 (2006) 783 [hep-th/0512149] [INSPIRE].
R. Blumenhagen, S. Moster and T. Weigand, Heterotic GUT and standard model vacua from simply connected Calabi-Yau manifolds, Nucl. Phys. B 751 (2006) 186 [hep-th/0603015] [INSPIRE].
R. Blumenhagen, S. Moster, R. Reinbacher and T. Weigand, Massless spectra of three generation U(N ) heterotic string vacua, JHEP 05 (2007) 041 [hep-th/0612039] [INSPIRE].
L.B. Anderson, Y.-H. He and A. Lukas, Heterotic compactification, an algorithmic approach, JHEP 07 (2007) 049 [hep-th/0702210] [INSPIRE].
L.B. Anderson, Y.-H. He and A. Lukas, Monad bundles in heterotic string compactifications, JHEP 07 (2008) 104 [arXiv:0805.2875] [INSPIRE].
L.B. Anderson, J. Gray, Y.-H. He and A. Lukas, Exploring positive monad bundles and a new heterotic standard model, JHEP 02 (2010) 054 [arXiv:0911.1569] [INSPIRE].
V. Braun, P. Candelas and R. Davies, A three-generation Calabi-Yau manifold with small Hodge numbers, Fortsch. Phys. 58 (2010) 467 [arXiv:0910.5464] [INSPIRE].
V. Braun, P. Candelas, R. Davies and R. Donagi, The MSSM spectrum from (0, 2)-deformations of the heterotic standard embedding, JHEP 05 (2012) 127 [arXiv:1112.1097] [INSPIRE].
L.B. Anderson, J. Gray, A. Lukas and E. Palti, Two hundred heterotic standard models on smooth Calabi-Yau threefolds, Phys. Rev. D 84 (2011) 106005 [arXiv:1106.4804] [INSPIRE].
L.B. Anderson, J. Gray, A. Lukas and E. Palti, Heterotic line bundle standard models, JHEP 06 (2012) 113 [arXiv:1202.1757] [INSPIRE].
L.B. Anderson, A. Constantin, J. Gray, A. Lukas and E. Palti, A comprehensive scan for heterotic SU(5) GUT models, JHEP 01 (2014) 047 [arXiv:1307.4787] [INSPIRE].
E.I. Buchbinder, A. Constantin and A. Lukas, The moduli space of heterotic line bundle models: a case study for the tetra-quadric, JHEP 03 (2014) 025 [arXiv:1311.1941] [INSPIRE].
Y.-H. He, S.-J. Lee, A. Lukas and C. Sun, Heterotic model building: 16 special manifolds, JHEP 06 (2014) 077 [arXiv:1309.0223] [INSPIRE].
E.I. Buchbinder, A. Constantin and A. Lukas, A heterotic standard model with B-L symmetry and a stable proton, JHEP 06 (2014) 100 [arXiv:1404.2767] [INSPIRE].
E.I. Buchbinder, A. Constantin and A. Lukas, Non-generic couplings in supersymmetric standard models, Phys. Lett. B 748 (2015) 251 [arXiv:1409.2412] [INSPIRE].
E.I. Buchbinder, A. Constantin and A. Lukas, Heterotic QCD axion, Phys. Rev. D 91 (2015) 046010 [arXiv:1412.8696] [INSPIRE].
A. Constantin, A. Lukas and C. Mishra, The family problem: hints from heterotic line bundle models, JHEP 03 (2016) 173 [arXiv:1509.02729] [INSPIRE].
A.P. Braun, C.R. Brodie and A. Lukas, Heterotic line bundle models on elliptically fibered Calabi-Yau three-folds, arXiv:1706.07688 [INSPIRE].
P. Candelas, Yukawa couplings between (2, 1) forms, Nucl. Phys. B 298 (1988) 458 [INSPIRE].
V. Braun, Y.-H. He and B.A. Ovrut, Yukawa couplings in heterotic standard models, JHEP 04 (2006) 019 [hep-th/0601204] [INSPIRE].
L.B. Anderson, J. Gray, D. Grayson, Y.-H. He and A. Lukas, Yukawa couplings in heterotic compactification, Commun. Math. Phys. 297 (2010) 95 [arXiv:0904.2186] [INSPIRE].
L.B. Anderson, J. Gray and B. Ovrut, Yukawa textures from heterotic stability walls, JHEP 05 (2010) 086 [arXiv:1001.2317] [INSPIRE].
S. Blesneag, E.I. Buchbinder, P. Candelas and A. Lukas, Holomorphic yukawa couplings in heterotic string theory, JHEP 01 (2016) 152 [arXiv:1512.05322] [INSPIRE].
S. Blesneag, E.I. Buchbinder and A. Lukas, Holomorphic Yukawa couplings for complete intersection Calabi-Yau manifolds, JHEP 01 (2017) 119 [arXiv:1607.03461] [INSPIRE].
E.I. Buchbinder, A. Constantin, J. Gray and A. Lukas, Yukawa unification in heterotic string theory, Phys. Rev. D 94 (2016) 046005 [arXiv:1606.04032] [INSPIRE].
P. Candelas and X. de la Ossa, Moduli space of Calabi-Yau manifolds, Nucl. Phys. B 355 (1991) 455 [INSPIRE].
P. Candelas, X. de la Ossa and J. McOrist, A metric for heterotic moduli, Commun. Math. Phys. 356 (2017) 567 [arXiv:1605.05256] [INSPIRE].
J. McOrist, On the effective field theory of heterotic vacua, Lett. Math. Phys. 108 (2018) 1031 [arXiv:1606.05221] [INSPIRE].
S.K. Donaldson, Scalar curvature and projective embeddings. I, J. Diff. Geom. 59 (2001) 479.
S. K. Donaldson, Scalar curvature and projective embeddings. II, Q. J. Math. 56 (2005) 345.
S.K. Donaldson, Some numerical results in complex differential geometry, math/0512625.
X. Wang, Canonical metrics on stable vector bundles, Comm. Anal. Geom. 13 (2005) 253.
M. Headrick and T. Wiseman, Numerical Ricci-flat metrics on K3, Class. Quant. Grav. 22 (2005) 4931 [hep-th/0506129] [INSPIRE].
M.R. Douglas, R.L. Karp, S. Lukic and R. Reinbacher, Numerical solution to the hermitian Yang-Mills equation on the Fermat quintic, JHEP 12 (2007) 083 [hep-th/0606261] [INSPIRE].
C. Doran et al., Numerical Kähler-Einstein metric on the third del Pezzo, Commun. Math. Phys. 282 (2008) 357 [hep-th/0703057] [INSPIRE].
M. Headrick and A. Nassar, Energy functionals for Calabi-Yau metrics, Adv. Theor. Math. Phys. 17 (2013) 867 [arXiv:0908.2635] [INSPIRE].
M.R. Douglas and S. Klevtsov, Black holes and balanced metrics, arXiv:0811.0367 [INSPIRE].
L.B. Anderson et al., Numerical Hermitian Yang-Mills connections and vector bundle stability in heterotic theories, JHEP 06 (2010) 107 [arXiv:1004.4399] [INSPIRE].
L.B. Anderson, V. Braun and B.A. Ovrut, Numerical Hermitian Yang-Mills connections and Kähler cone substructure, JHEP 01 (2012) 014 [arXiv:1103.3041] [INSPIRE].
J.J. Heckman and C. Vafa, Flavor hierarchy from F-theory, Nucl. Phys. B 837 (2010) 137 [arXiv:0811.2417] [INSPIRE].
A. Font and L.E. Ibáñez, Matter wave functions and Yukawa couplings in F-theory Grand Unification, JHEP 09 (2009) 036 [arXiv:0907.4895] [INSPIRE].
J.P. Conlon and E. Palti, Aspects of flavour and supersymmetry in F-theory GUTs, JHEP 01 (2010) 029 [arXiv:0910.2413] [INSPIRE].
L. Aparicio, A. Font, L.E. Ibáñez and F. Marchesano, Flux and instanton effects in local F-theory models and hierarchical fermion masses, JHEP 08 (2011) 152 [arXiv:1104.2609] [INSPIRE].
E. Palti, Wavefunctions and the point of E 8 in F-theory, JHEP 07 (2012) 065 [arXiv:1203.4490] [INSPIRE].
A. Lukas, B.A. Ovrut and D. Waldram, Boundary inflation, Phys. Rev. D 61 (2000) 023506 [hep-th/9902071] [INSPIRE].
A. Lukas, B.A. Ovrut and D. Waldram, On the four-dimensional effective action of strongly coupled heterotic string theory, Nucl. Phys. B 532 (1998) 43 [hep-th/9710208] [INSPIRE].
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Blesneag, Ş., Buchbinder, E.I., Constantin, A. et al. Matter field Kähler metric in heterotic string theory from localisation. J. High Energ. Phys. 2018, 139 (2018). https://doi.org/10.1007/JHEP04(2018)139
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DOI: https://doi.org/10.1007/JHEP04(2018)139